etermine the focus and directrix of the parabola with the given equation. Sketch the graph, and indicate the focus, directrix, and vertex if the equation is y^2=20x

To determine the focus and directrix of the parabola given by the equation \( y^2 = 20x \), let's go through the following steps:

### 1. **Identify the Standard Form:**
The equation \( y^2 = 20x \) resembles the standard form of a parabola that opens either to the right or to the left, which is given by:

\[
y^2 = 4px
\]

### 2. **Find the Value of \( p \):**
By comparing the given equation \( y^2 = 20x \) with the standard form \( y^2 = 4px \), we can identify that \( 4p = 20 \). To find \( p \), divide both sides by 4:

\[
p = \frac{20}{4} = 5
\]

### 3. **Determine the Vertex, Focus, and Directrix:**

- **Vertex:** Since the equation is in the form \( y^2 = 4px \), and there is no vertical or horizontal shift, the vertex is at the origin, \( (0, 0) \).
- **Focus:** The focus lies at \( (p, 0) \), which is \( (5, 0) \).
- **Directrix:** The directrix is a vertical line given by \( x = -p \), which is \( x = -5 \).

### 4. **Sketch the Graph:**
To sketch the graph:
- Draw the parabola opening to the right with the vertex at the origin.
- Mark the focus at \( (5, 0) \).
- Draw the directrix as a vertical line at \( x = -5 \).

Here's a textual description of the graph:
- The parabola opens to the right from the vertex at \( (0, 0) \).
- The focus \( (5, 0) \) lies on the x-axis, 5 units to the right of the vertex.
- The directrix \( x = -5 \) is a vertical line 5 units to the left of the vertex.

### Summary:
- **Vertex:** \( (0, 0) \)
- **Focus:** \( (5, 0) \)
- **Directrix:** \( x = -5 \)

If you need the visual sketch or have any questions, feel free to ask!

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Would you like to see more details or have any other questions? Here are some related topics you might find interesting:

1. How to convert a parabolic equation to vertex form.
2. Understanding the relationship between the focus and directrix in different parabola orientations.
3. How to find the equation of a parabola given its vertex and focus.
4. Analyzing parabolas that open upwards or downwards instead of sideways.
5. The significance of the parameter \( p \) in the context of a parabola's properties.

**Tip:** When analyzing parabolas, remember that the distance from any point on the parabola to the focus equals the distance from that point to the directrix!

Learn how to determine the focus and directrix of the parabola y^2 = 20x with a detailed explanation. This problem covers the vertex form of a parabola, focusing on identifying key properties such as the vertex, focus, and directrix. Suitable for high school students.

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